# Trial and Improvement

## solving equations using trial and improvement

Trial and Improvement is a method of solving equations when you can't do it by normal algebraic methods. It's normally a 3 mark question so understanding it is vital if your aiming for a grade C in maths! You will need a calculator to answer these questions.

## Example Question

a) Show that the equation x3 - 6x + 1 = 0 has a solution between 2 and 3.
b) Solve this equation correct to 1 decimal place.

## Solution

a) To show that x3 - 6x + 1 = 0 has a solution between 2 and 3 we need to substitute x = 2 and x = 3 into the equation.

When x = 2 we get: (2)3 - 6(2) + 1 = 8 - 12 + 1 = -3.
When x = 3 we get: (3)3 - 6(3) + 1 = 27 - 6(3) + 1 = 10.

Now notice when x = 2 we get an answer less than 0 and when x = 3 we get an answer greater than 0.  This means there must be a value of x between 2 and 3 which is equal to 0.

b) We now need to find this solution correct to 1 decimal place. To do this we need to draw a table to test different values of x. Since we know the answer is between 2 and 3 it makes sense to start with x = 2.5.

x x3 - 6x + 1 Comment
2 -3 Too small
3 10 Too big
2.5 1.625 Too big
2.3 -0.633 Too small
2.4 0.424 Too big

Now since 2.3 was too small and 2.4 was too big we know the solution is between these values.  To find our answer to 1 decimal place we have to try one more value in the middle of these, when x = 2.35.

When x = 2.35 we get: (2.35)3 - 6(2.35) + 1 = -0.122125.

Finally since 2.35 is too small then we can say 2.3 must also be too small. Hence x = 2.4 to 1 decimal place.

## Test Yourself!

The equation x3 + x = 15 has a solution between 2 and 3.
Find this solution correct to one decimal place.

x = The equation x3 - 2x + 6 = 0 has a solution between -3 and -2.
Find this solution correct to one decimal place.
x = ### Interactive worksheets on sequences:

Trial and improvement worksheet